SAN FRANCISCO—The most famous court reporter in the country, Mark Kislingbury, calmly rested his hands on his keyboard. Around him, more than three dozen other stenographers also readied themselves for the 2014 national court-reporting championships, held last month in a hotel conference room here.

The "Olympics of court reporting," as it is often called, kicked off with a speech, read at precisely 200 words a minute. The stenographers' fingers began to fly. Onlookers, numbering fewer than the contestants,...